All Questions
54 questions
69
votes
2
answers
4k
views
Function that produces primes
For any $n\geq 2$ consider the recursion
\begin{align*}
a(0,n)&=n;\\
a(m,n)&=a(m-1,n)+\operatorname{gcd}(a(m-1,n),n-m),\qquad m\geq 1.
\end{align*}
I conjecture that $a(n-1,n)$ is always ...
- 4,938
56
votes
1
answer
4k
views
A mysterious connection between primes and $\pi$
The Prime Number Theorem relates primes to the important constant $e$.
Here I report my following surprising discovery which relates primes to $\pi$.
Conjecture (December 15, 2019). Let $s(n)$ be ...
- 15.6k
28
votes
0
answers
716
views
Does this infinite primes snake-product converge?
This re-asks a question I posed on MSE:
Q. Does this infinite product converge?
$$
\frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \...
- 151k
24
votes
1
answer
2k
views
Parity of the multiplicative order of 2 modulo p
Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
- 429
16
votes
1
answer
4k
views
Order of magnitude of $\sum \frac{1}{\log{p}}$
Question: What is the order of magnitude of the following sum?
$$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{\substack{p<n\\\text{...
- 3,004
16
votes
1
answer
1k
views
Divergence of a series similar to $\sum\frac{1}{p}$
Suppose we start with $k$ primes $p_1,p_2,\ldots ,p_k$ (not necessarily consecutive) and a residue class for each prime $r_1,r_2,\ldots ,r_k$.
We denote the least integer not covered by the arithmetic ...
- 3,866
10
votes
1
answer
694
views
Prime numbers from permutation
Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and reverse-cyclically permuting every $n$ consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $...
- 4,938
8
votes
2
answers
3k
views
Asymptotics of the least common multiple of the first natural numbers
What is $$ \limsup_{n \to \infty} \frac{\log(\mathrm{lcm}(1,2, \dots, n))}{n} \ \ ?$$
- 11.3k
8
votes
2
answers
756
views
The sequence $a_{n+1}=$ the greatest prime factor of $(xa_n+y)$
Let $\operatorname{ GPF}(n)$ be the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.
Is there a way to prove that the sequence $a_{n+1}=\operatorname{ GPF}(...
- 83
8
votes
0
answers
1k
views
On the sum of consecutive primes and product of first and last
Lets consider the sequence of consecutive prime numbers $p_1=2 , p_2=3 ,p_4=5 , \cdots$
. $(p_n,p_{j})$ is to be called good prime pair if $$\sum_{i =n }^{j}p_i= p_n p_{j}$$
Meaning the sum of set of ...
- 305
7
votes
2
answers
606
views
convergence in $\hat{\mathbb{Z}}$, modulo prime power
The following problem appears in Lenstra's Galois Theory for Schemes (p 14, Ex 1.16).
Let $b\in\mathbb Z_{\ge0}$. Define the sequence $(a_n)_{n=0}^\infty$
by $a_0=b, a_{n+1}=2^{a_n}$. Prove that $...
- 101
7
votes
1
answer
497
views
Questions involving primes $p\equiv1\pmod4$
As claimed by Fermat and proved by Euler, any prime $p\equiv1\pmod4$ can be written uniquely as $s_p^2+t_p^2$ with $s_p,t_p\in\{1,2,3,\ldots\}$, $2\nmid s_p$ and $2\mid t_p$. For any positive integer $...
- 15.6k
6
votes
3
answers
1k
views
Does this 'alternating' Euler product converge for all $\Re(s) > 0$?
Does the following 'alternating' Euler product, with $p_n$ the $n$-th prime number, converge for $\Re(s)>0$ ?
$$\displaystyle \prod_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p_{n}^{s}}} \right)^{(...
- 4,169
6
votes
3
answers
822
views
Is it possible to multiply two series to get as a result all composite numbers?
I was toying with the following problem:
Is it possible to find two infinite integer sequences $(a_n), (b_n)>0$ such that $\sum_{n=1}^{\infty}\frac{1}{(a_n)^s}\cdot \sum_{n=1}^{\infty}\frac{1}{(b_n)...
- 3,866
6
votes
1
answer
279
views
Which $n$ have $\lvert\{2^n-2^k -1\}\cap {\mathrm{PRIMES}}\rvert=m$?
Consider numbers of the form $2^n - 2^k - 1$ with $k < n$ as considered in OEIS sequence A208083. As for A208083 I investigated how many of these numbers are prime, but turned the question around: ...
- 9,720
6
votes
1
answer
393
views
Test for pair of odd primes $(p, 2p^2-1)$
Let $a(n)$ be A106483 (i.e., primes $p$ such that $2p^2-1$ is also prime).
Let $b(n)$ be an integer sequence such that $b(n) = B$ after the whole transformation where we start with $A = n$, $B = 1$, $...
- 4,938
6
votes
0
answers
257
views
Convergence with the recurrence $T_{n+1}=T_n^2-T_n+\frac{n}{p_n}$
For each integer $n\geq 1$ I define the recurrence $$T_{n+1}=T_n^2-T_n+\frac{n}{p_n},$$
with $T_1=1$, where $p_k$ denotes the $k$-th prime.
So multiplying by $(-1)^n$ and telescoping gives that for ...
5
votes
2
answers
529
views
Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?
OEIS A226181:
3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79,
83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ...
Primes $p$ ...
- 359
5
votes
1
answer
287
views
Is the factorization of $a_m-a_n$ affected by the fact that $\Sigma \frac{1}{a_k}<+\infty$?
I would like to ask the following.
Let $(a_n)$ be a sequence of natural numbers such that
$\sum_{k=1}^{\infty}\frac{1}{a_k}$ converges. Is it true that for
infinitely many $m$, there is a $n<m$ ...
- 3,866
5
votes
1
answer
943
views
Are all the numbers $\pi(n^2)/n^2\ (n=1,2,3,\ldots)$ pairwise distinct?
For $x>0$ let $\pi(x)$ denote the number of primes not exceeding $x$. A well-known conjecture of Legendre states that $\pi(n^2)<\pi((n+1)^2)$ for any positive integer $n$. Here I ask the ...
- 15.6k
4
votes
2
answers
650
views
Conjectured relation between alternating Prime zeta series and Riemann zeta
Let $P(s)$ be the Prime zeta function.
Numerical evidence suggests these identities:
$$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad (1)$...
- 25.4k
4
votes
1
answer
628
views
Asymptotics for $\prod(1-\frac{1}{p})$ over all primes $p\leq x$ with $p \equiv 3 \bmod 4$
Let us define the following functions:
\begin{equation*}
\small A(x)=\prod_{\substack{p\leq x\\ p\equiv 3 \bmod 4}} \Big(1-\frac{1}{p}\Big), \mbox{ } \mbox{ }
B(x)=\prod_{\substack{p\leq x\\ p\...
- 3,098
4
votes
1
answer
441
views
Two conjectural infinite series for $\pi$
I am looking for a proofs of the following two claims:
Claim 1.
$$\frac{2\pi}{\sqrt{3}}=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{\Omega_1(n)}}{n}$$ where $\Omega_1(n)$ is the number of prime ...
- 2,661
4
votes
1
answer
206
views
Prime numbers and number of partitions of $n$ into distinct parts with boundary size $2$
Let $a(n)$ be A227559, i.e., number of partitions of $n$ into distinct parts with boundary size $2$. Be careful here: offset is $3$.
I conjecture that $a(4n+2)=2n+1$ for $n>0$ if and only if $2n+1$ ...
- 4,938
4
votes
2
answers
891
views
Asymptotics for the number of ways to sum primes such that the sum is <= n
Hello!
Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p_1 + \ldots + p_k \leq n \quad (1) $$ where $k$ is arbitrary and $p_1 \leq \...
- 3,463
4
votes
1
answer
219
views
Numbers $n$ whose representation as the product of two divisors require more digits than that of $n$
Note: Posting in MO since it was unanswered in MSE
Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > ...
- 5,470
4
votes
1
answer
306
views
Can we always attain another prime via inserting digits between the digits of a fixed prime?
The sequence OEIS A080437 is
For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m.
I don't see why this ...
- 25.4k
3
votes
1
answer
323
views
Name and properties of $\mathrm{lcm}(\{1,\,\cdots,\,n\})$ [closed]
one of the most prominent functions of the first $n$ natural numbers is the factorial $n!$ that denotes their product.
Today however I wondered whether the least common multiple $\mathrm{lcm}(n):=\...
- 13.2k
3
votes
1
answer
188
views
About the sum $S(p_n)=\sum_{1\le k\lt n}\,p_n\mod\;p_k$
For $\,p_n\gt2\,$ let's define the sum $\,S(p_n)=\sum_{1\le k\lt n}\,p_n\;mod\;p_k$, where $\,p_k\,$ represents the $\,k$-th prime.
The first terms of the sequence $\,S(p_n)\,$ (OEIS A033955 - sum of ...
- 1,008
3
votes
0
answers
330
views
Can you prove and/or generalize this formula involving the Möbius function at n = square free numbers for elliptic curve related sequence in the OEIS?
Let $g(n)$ be the Dirichlet inverse of the Euler totient function:
$$g(n) = \sum\limits_{d|n} d \cdot \mu(d)$$
and let $f(x,y)$ be the elliptic equation:
$$f(x,y)=x^3 - x^2 - y^2 - y$$
Show that the ...
- 1,183
3
votes
0
answers
183
views
From Firoozbakht's conjecture to set interesting conjectures for sequences or series of primes
In this post we denote the $k-th$ prime number as $p_k$. I present two conjectures, the first about the asymptotic behaviour of a product and the other about an alternating series. I don't know if ...
2
votes
1
answer
292
views
Difference between $n$-th and $(n-1)$-th composite numbers
Let $f(n)$ = 1 if $n$ belongs to A014689, $\operatorname{prime}(n)-n$, the number of nonprimes less than $\operatorname{prime}(n)$. Here $\operatorname{prime}(n)$ is the $n$-th prime number, $\...
- 4,938
2
votes
1
answer
401
views
Is a certain sumset derived from primes of a certain form the set of all naturals?
OEIS A167055 Numbers n such that $12n + 5$ is prime.
$0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS $A167055$.
I conjecture that the set of the sum of every two items of this ...
- 31
2
votes
1
answer
284
views
Question on odd positive integers and Fermat factors
Let $n$ be an odd positive integer, Let $o=\operatorname{ord}_n 2$ be the order of 2 modulo $n$ and $m$ the period of $1/n, k$ is number of distinct odd residues contained in set $\{2^1,2^2,...,2^{n−1}...
- 359
2
votes
1
answer
154
views
GCD for two Cullen numbers
The $n$'th Cullen number is $C_n = n\cdot2^n+1$.
If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?
- 21
2
votes
1
answer
354
views
Formulas for $\sum_{x=1}^{n}\Big\{\frac{x^q}{n}\Big\}$
Consider sum:
\begin{equation}
S_q(n) = \sum_{x=1}^{n}\Big\{ \frac{x^q}{n} \Big\}
\end{equation}
where $\{x\}$ is fractional part of $x$. It's easy to see that $S_{1}(n) = \frac{1}{2}(n-1)$, but ...
- 267
2
votes
0
answers
131
views
Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?
I am trying to get an asymptotic formula such as
$$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$
where $L_4(s, n)$ is the first $n$...
- 3,098
2
votes
0
answers
199
views
Not a twin prime pair test using $\gcd$ only
Let $m$ be an odd positive integer such that $m=2k+1$, $k\in\mathbb{N}$.
Let $v$ be a vector of $n$ positive integers. Let $v(i)$ be the $i$-th element of the vector. Then we start with $v(i)=m(i+1)-2$...
- 4,938
2
votes
0
answers
215
views
Two conjectures about generalised A329369
Let $m \geqslant 2$ be a fixed integer.
Let
$$\operatorname{wt}(n,m)=\operatorname{wt}\left(\left\lfloor\frac{n}{m}\right\rfloor,m\right)+n\bmod m, \operatorname{wt}(0,m)=0$$
Then we have an integer ...
- 4,938
2
votes
0
answers
422
views
Sequences with high densities of primes: how to boost them to get even more and larger primes
I propose a methodology to help find large prime numbers with a much higher probability than picking up random numbers and testing them for primality. This would help speed up prime number generators ...
- 3,098
2
votes
0
answers
156
views
Questions about a certain sequence of naturals generated by primorials
I'm working on the following sequence of naturals (which is NOT listed in OEIS)
$$3,5,11,17,23,29,59,89,119,149,179,209,419,629,839,1049,1259,1469,1679,...$$
whose elements are generated this way
$$3=(...
- 1,008
2
votes
0
answers
131
views
How many divisors of $\phi(m)$ do not divide $m-1$?
Lehmer's totient problem asks if there exists a composite number $m$ such that $\phi(m)$ divides $m-1$. Lower bounds on $m$ has been established but we do not know if a solution exists. Clearly, if we ...
- 5,470
2
votes
0
answers
167
views
What about series involving strong primes?
I know about the importance in analytic number theory of the sutdy of series involving prime numbers or constellations of prime numbers, for example, if I am not wrong, major theorems are Mertens' ...
2
votes
0
answers
306
views
Conjectured initial values of Inkeri's primality test for Fermat numbers
This is a repost of this question .
Can you provide a proof or a counterexample to the claim given below ?
First , we shall give a definition of the Inkeri's primality test for Fermat numbers :
...
- 2,661
1
vote
1
answer
123
views
Periodic sequences of integers generated by $a_{n+1}=\frac{\operatorname{rad}(pa_{n})}{p}+\frac{\operatorname{rad}(qa_{n-1})}{q}$
Let's define the radical of the positive integer $n$ as
$$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p$$
and consider the sequence
$$a_{n+1}=\frac{\operatorname{rad}(p\cdot a_{n})...
- 1,008
1
vote
0
answers
89
views
Test for odd prime triples in a $2p-1$ progression
Let $a(n)$ be A057326 (i.e., first member of a prime triple in a $2p-1$ progression).
Let $b(n) = B$ after $n-1$ iterations where we start with $A=n, B=1$ and for $i$ from $1$ to $n-1$ simultaneously ...
- 4,938
1
vote
0
answers
148
views
About the distribution of Fibonacci numbers that are primes
Let's consider the Fibonacci sequence, that is the sequence of naturals defined by:
$F_1=F_2=1$
$F_{n+1}=F_{n}+F_{n-1}$
It is an open problem whether the sequence contains an infinite number of ...
- 1,008
1
vote
0
answers
200
views
Limit of Sequence of unusual Prime Product
Let $p_n$ be the nth prime and $p_L$ be closest to its square root:
\begin{equation}
p_L^2 \approx p_n \approx x
\end{equation}
Let $\sigma \in Z^+$ be a positive integer constant. Define the ...
0
votes
1
answer
251
views
Series of reciprocals of smooth numbers
Be a non-empty set of primes $A $. Let us define $A^{\otimes}$ as the set of numbers smooth over $A$, that are the naturals having all their prime divisors in $A$ (where $1$ is arbitrarily considered ...
- 113
0
votes
1
answer
100
views
$a(16n+k)=b(16n+k)-c(16n)$ for $n\geqslant0$, $0 < k < 16$ where $c(n)=b(n)-a(n)$
Let $a(n)$ be A339970 = A329697$($A019565$(2n))$: the sequence begins with
$$0, 1, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 5, 6, 2, 3, 3, 4, 4$$
Also let's consider
$$\ell(n)=\left\lfloor\log_{2}(n)\right\...
- 4,938